2017 Mathcounts State Prep: Volume of a Regular Tetrahedron and Its Relationship with the Cube it's Embedded

How to find volume of a tetrahedron (right pyramid) with side length one.

The above link gives you a visual interpretation of the relationship of a regular tetrahedron, its
relationship with the cube that it is embedded and the other kind of tetrahedron (right angle pyramid).

The side of the cube is ${\displaystyle \frac{S}{\sqrt{2}}}$ so the volume of the regular embedded tetrahedron is
${\displaystyle \frac{1}{3}}\times {\left({\displaystyle \frac{S}{\sqrt{2}}}\right)}^3$=${\displaystyle \frac{1}{3}}\ast {\displaystyle \frac{s^3}{2\sqrt{2}}}$= ${\displaystyle \frac{\sqrt{2}{S}^3}{12}}$.

You can also fine the height of the tetrahedron and then ${\displaystyle \frac{1}{3}}$*base*space height to get the volume.
Using Pythagorean theory, the hypotenuse S and one leg which is ${\displaystyle \frac{2}{3}}$ of the height of the equilateral triangle base, you'll get the space height.